On the Stability of Equilibrium for Periodic Mechanical Systems

Trends in the Theory and Practice of Non-Linear Analysis

V. Lakshmikantham (Editor)

309

Elsevier Science Publishers B.V. (North-Holland), 1985

ON THE STABILITY OF EQUILIBRIUM FOR PERIODIC MECHANICAL SYSTEMS Vinicio Moauro Dipartimento di Matematica Universit; di Trento Povo, Trento ITALY Some results, obtained in C1,21 , relative to the stability properties of an isolated equilibrium position of a holonomic mechanical system are presented. Here the proofs will be only sketched.

3

Let be a holonomic mechanical system with n 1 degrees of freedom and let q , , ...,q,, be a system of Lagrangian coordinates f o r g . Denoting by q the n-vector (4,,...,q_ ) , we will suppose that

Gh 6;

1 ”

(h,) the kinetic energy of 5 has the f o r m y = 7 (h,) the forces acting on 3 are the following: 1 ) a force depending on a potential function U(q,t) f(t)U*(q), where f(t) is a periodic function of the time t,with period T > 0, not identically equal to zero, and U*(q) is a ?-function defined in a neighbourhood of q = 0 which is not flat 0. Further we assume that the minimum order of thenan at q zero derivatives of U * a t q = 0 is m+l with m 2 and we will denote by U t + , the homogeneous polynomial in q o f degree m+l such that U*(q) = Utrr,+o ( ~q~””‘),with 1 . 1 any Euclidean norm in R“; a linear complete dissipative force with Lagrangian compo-th; nents Q h = 0 is an isolated equilibrium position f o r s . The stability problem of the equilibrium position q = 0 o f 3 letely solved in the case f(t) 1. In fact, in such a case, q I 0 is asymptotically stable if U * has a maximum at q = 0 and it is unstable if U* has not a maximum at q 0. This result is true also when the kinetic energy is any positive definite quadratic form in (i with coefficients depending on q , U * is flat and the complete dissipative force is not linear. The equations of motion of dt

3

are the following: h = l...,n.

!l Moauro

310

CASE n

1.

In [ 1 1 the stability problem of the null solution of ( 1 ) (that is of the equilibrium position q = 0 of 3 ) has been considered, in connection to the problem of generalized Hopf bifurcation for periodic systems, in the case n = 1. In this case the linear approximation of ( 1 ) has 0 as simple characteristic exponent. Therefore, by using also a procedure due to Liapunov C31, it is possible to get the following results:

(A) If f(t) has mean value equal to zero, then the null solution of ( 1 ) is (2m-l)-asymptotically stable; (B) If f(t) has mean value M f

0, then the null solution of ( 1 ) is m-asymptotically stable or m-unstable whether MULT,is negative definite or not.

Remarks. For any integer k 1 , saying that the null solution of(l) is k-asymptotically stable or k-unstable means that its asymptotic stability or its instability is determined only by the terms of order less than or equal to k in the expansion of the right hand sides of ( 1 ) and k is the minimum integer for which this happens. Result ( B ) shows that, if f(t)= 1, the null solution of ( 1 ) is m-asymptotically stable or m-unstable whether U$,.,has or has not a maximum at q = 0. Thus, our hypotheses (h,), (h,) assure that the stability properties can be recognized at the order m. Result ( A ) shows that the introduction in the potentialfunction of a periodic factor with mean value equal to zero can change drastically the stability properties of the equilibrium position. In the one degree of freedom case, the property of the equilibrium position to be isolated is assured by the non flatness of U * and it is recognizable by inspecting U+,+,only. The procedure to get results following two steps: STEP 1. Look f o r a change of into a system of the form k = gx + i = -Y +

x

(A) and (B) can be schematized in the coordinates which trasforms system ( 1 ) X(x,y,t) Y(X,Y,t)

where k is an integer greater than or equal to 2, g $ 0 is a constant, X,Y are functions of order greater than or equal to 2 in x , y at x = y 0, T-periodic in t and such that X(x,O,t), Y(x,O,t) are of order greater than k in x at x = 0. This change of coordinates is required to preserve the stability properties of the null solution and the order at which such properties are recognized. STEP 2. Construct a Liapunov function, by using only the terms up to the order k in the r.h.s. of ( 2 1 , which assures the asymptotic stability or the instability of the null solution of (2). We do not give the details of this procedure because in the following we will show that it can be generalized to the case n 2 1. Therefore we will give the details in this general case.

311

The Stability for Mechanical Systems CASE n 21.

By f o l l o w i n g t h e s c h e m e o f t h e p r o c e d u r e u s e d i n t h e c a s e n 1, it i s p o s s i b l e t o show [ 2 1 t h a t s y s t e m ( 1 ) c a n b e t r a n s f o r m e d , by u s i n g c h a n g e s o f c o o r d i n a t e s which do n o t modify t h e s t a b i l i t y p r o p e r t i e s o f t h e n u l l s o l u t i o n , i n t o a system o f t h e form

-

(3)

. ..

. ..

w h e r e I-, = (q, ,I-,,), 5 = (c,, ,k ) , t h e f u n c t i o n s g, ( I - , ) a r e homogen e o u s p o l y n o m i a l s o f same d e g r e e k 2 2 , V h , W, a r e f u n c t i o n s of o r d e r g r e a t e r t h a n o r e q u a l t o 2 i n 1-,,5 a t q = 5 = 0 , T - p e r i o d i c i n t and s u c h t h a t n h ( q , O , t ) , h ( ~ - , , O , t )a r e i n f i n i t e s i m a l o f o r d e r g r e a t e r t h a n k i n 0 a t I-, 0; - under t h e hypoyhesis

(H)

Igradu&l12 is positive definite, i.e. the property of the equilib r i u m p o s i t i o n q = 0 t o b e i s o l a t e d i s r e c o g n i z a b l e by i n s p e c t i n g o n l y Uk.,,

a L i a p u n o v f u n c t i o n c a n b e c o n s t r u c t e d , by u s i n g o n l y t h e terms o f o r d e r less t h a n o r e q u a l t o k i n t h e r . h . s . o f ( 3 1 , s o t h a t asymptot i c s t a b i l i t y o r i n s t a b i l i t y of t h e n u l l s o l u t i o n of ( 3 ) is assured. To r e d u c e ( 1 ) t o a s y s t e m o f f o r m ( 3 1 , l e t u s i n t r o d u c e f i r s t t h e new v a r i a b l e s d e f i n e d by Xh

= qh

+

qh

System ( 1 ) is t r a n s f o r m e d i n t o

(4)

kh

= f(tA-

aU*

;lh

yh

h = I,

h x

)q:r-y

...,n .

I , ...,n , ( x , ,...,x , ) ,

h

We c a n c o n s t r u c t a s i n [ 1 , 4 1 n p o l y n o m i a l s Q h = I , ...,n , i n x o f b’ d e g r e e 2m-I, w h o s e c o e f f i c i e n t s a r e T - p e r i o d i c f u n c t i o n s o f t , s u c h t h a t a l o n g t h e s o l u t i o n s o f ( 4 ) we have

We h a v e

h = I , ...,n , 8 ( x , t ) + O(Ixlm) h h,m homogeneous p o l y n o m i a l s i n x o f d e g r e e m s a t i s f y i n g t h e d (x,t)

with 0 b m c o n d it i b n (5)

By m e a n s o f t h e p o l y n o m i a l s a h ’ s , w e d e f i n e t h e new v a r i a b l e s x

Gh = x h h and ( 4 ) i s t r a n s f o r m e d i n t o

= Yh - Q h ( X , t )

h = I,

...,n ,

3 12

V. Moauro

...,

where f o r h I, n , X ,d are f u n c t i o n s of o r d e r greater than o r h h e q u a l t o m i n (x,c) a t x = 5 = 0 , T - p e r i o d i c i n t and X ( x , O , t ) , h d ( x , O , t ) a r e o f o r d e r g r e a t e r t h a n 2m-1 i n x a t x = 0 . We w i l l d i s k i n g u i s h now t h e t w o s i t u a t i o n s : ( I ) f ( t ) h a s mean v a l u e e q u a l t o z e r o ; ( 1 1 ) f ( t ) h a s mean v a l u e M d i f f e r e n t f r o m z e r o . ( I ) By m e a n s o f c h a n g e s o f c o o r d i n a t e s , u s e d i n t h e a v e r a g i n g m e t h o d [ 5 , 6 1 , we c a n t r a n s f o r m s y s t e m ( 6 ) i n t o t h e f o l l o w i n g s y s t e m :

w h i c h has t h e f o r m ( 3 ) w i t h t h e g I s h o m o g e n e o u s p o l y n o m i a l s o f d e h g r e e 2m-1. (11) By a v e r a g i n g d i r e c t l y s y s t e m ( 6 ) , we g e t t h e s y s t e m

-sh

Sh

+

Wh(Il,3,t)

w h i c h h a s a l s o t h e f o r m ( 3 ) w i t h t h e g I s homogeneous p o l y n o m i a l s o f h degree m. Now w e w i l l show how i t i s p o s s i b l e t o c o n s t r u c t L i a p u n o v f u n c t i o n s f o r s y s t e m s ( 7 ) a n d ( 8 ) i n s u c h a way t h a t w e c a n p r o v e t h e f o l l o w ing THEOREM. Under h y p o t h e s i s (H), r e s u l t s ( A ) a n d ( B ) h o l d t r u e i n t h e case n > 1. In s i t u a t i o n ( I ) let us consider the function

=

" 1 F h [ T (Ilk

+ 5;)

+
w h e r e t h e P ' s a r e p o l y n o m i a l s i n q o f d e g r e e 2m-1 a n d i n f i n i t e s i m a l h o f o r d e r g r e a t e r t h a n o r e q u a l t o m a t rl = 0 , w h o s e c o e f f i c i e n t s a r e T-periodic f u n c t i o n s of t . These c o e f f i c i e n t s can be chosen i n such a way t h a t t h e d e r i v a t i v e o f V a l o n g t h e s o l u t i o n s o f s y s t e m ( 7 ) i s g i v e n by with (9) where v ; L,, ' v>,v a r e c o n t i n u o u s f u n c t i o n s o f n , G , t , ? - p e r i o d i c i n t a n d ' i n f i n i t e s i m a l i n 0 , a ~t q 5 0 . The f u n c t i o n V i s n e g a t i v e d e f i n i t e b e c a u s e , by ( 5 ) a n d h y p o t h e s i s ( H ) , w e h a v e f o r Q # 0 I

313

The Stability for Mechanical Systems

Thus, the null solution of (7) is asymptotically stable and it is (2m-l)-asymptotically stable because the determination of the polynomials P I s in V depends only on the terms of order less than or equal to 3m-I in the r.h.s. of ( 7 ) . In situation (II), let us suppose first that MU* m+ 1 is not sign-constant in a neighbourhood of q = 0. In such a case, let us consider the function where the P h q s are homogeneous polynomials of degree 2m-1 in q whose coefficients are T-periodic functions of t determined in such a way that along the solutions of system (8) we have v -X [M2-(-auk*, + 51; + S ( l l , 5 , t) ,h T aqh ' 9 ' 7 with s a function defined as in (9). Thus, by hypothesis (HI, v is negative definite and the null solution of (8) is unstable. As the determination of the polynomials P I s in V depends only on the terms h of order less than or equal to m in the r.h.s. of ( a ) , the null solution of (8) is m-unstable. Let us suppose now that MU* is sign m+ 1 constant in a neighbourhood of q 0. Because of hypothesis (HI, MU*, will be sign definite. If it is negative definite, we consider $he function

,

v = i_ h [ 1 2 ( q; + 5 ' ) + 5 h Ph (ll,t)l, where the P I s are hbmogeneous poPynomials of degree m in q whose h coefficients are T-periodic functions of t determined in such a way that along the solutions of (8) with

s =F--".+Lk'm ;. *

' ' . I . . .

I

in

.:71

. .;l

n

+

'si- ; s 5 , 5 ,

,

,IT=+

where ii,,,, . , , L,,,"C,,v are continuous functions of q , ~ , t, T-periodic ~ q 5 = 0. Therefore the null soluin t and infinitesimal in q , at tion of (8) is asymptotically stable and the way of constructing the polynomials P ' s implies that such property is recognizable at the is positive definite, it is possible to order m. FinaPly, if MU* prove that the null solu!ion tl of (8) is m-unstable by using a function of the type with the

Ph * s

suitable homogeneous polynomials in

tl

of degree m.

Remark. As a consequence of result (B), the equilibrium position q = 0 of 3 is m-unstable if f(t) has mean value different from zero and m is even. REFERENCES: [l]

Salvadori,L., Visentin,F., Sul problema della biforcazione generalizzata di Hopf per sistemi periodici, Rend. Sem. Mat. Univ. di Padova 68 (1982).

V. Moauro

314

A stability problem for holonomic mechanical systems, Ann. Mat. Pura e Appl., to appear.

[ 2 1 Moauro,V.,

[ 3 1 Liapunov,A.M., Problbme g6n6ral de la stabilit6 du mouvement, Ann. of Math. Studies 17 (Princeton Univ. Press, New Jersey 1947).

[4] Bernfeld,S.R., Negrini,P., Salvadori,L., Quasi invariant manifolds, stability and generalized Hopf bifurcation, Ann. Mat. Pura e Appl. (IV) 130 (1982). [ 5 ] Hale,J.K., Ordinary differential equations (Wiley Interscience, New York, 1969).

[ 6 1 Ladis,N.N., Asymptotic behaviour of solutions of quasi homogeneous differential systems, Differential Equations 9 (1973).

The detailed version of this paper has been submitted for publication elsewhere.